On the stability of solutions to random optimization problems under small perturbations

TL;DR

This paper demonstrates that solutions to certain random optimization problems, like the Euclidean TSP, remain stable under small perturbations, with the number of distinct optimal solutions remaining bounded as problem size grows.

Contribution

It introduces a general stability framework applicable to various models, showing solutions are not highly sensitive to small changes.

Findings

Number of distinct optimal paths in Euclidean TSP is bounded as n→∞.

Stability results extend to models like spin glasses and random matrices.

Solutions exhibit robustness under small perturbations across multiple settings.

Abstract

Consider the Euclidean traveling salesman problem with $n$ random points on the plane. Suppose that one of the points is shifted to a new random location. This gives us a new optimal path. Consider such shifts for each of the $n$ points. Do we get $n$ very different optimal paths? In this article, we show that this is not the case - in fact, the number of truly different paths can be at most $\mathcal{O}(1)$ as $n\to \infty$. The proof is based on a general argument which allows us to prove similar stability results in a number of other settings, such as branching random walk, the Sherrington-Kirkpatrick model of mean-field spin glasses, the Edwards-Anderson model of short-range spin glasses, and the Wigner ensemble of random matrices.